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Recent Experience With Large Scale Urban Development Models
Copyright © IFAC Theory and Application of Digital Control New Delhi . India 1982
RECENT EXPERIENCE WITH LARGE SCALE URBAN DEVELOPMENT MODELS B. G. Hutchinson and H. M. Couclelis Department of Civil Engineering, University of Waterloo , Waterloo, Ontario N2L 3Gl, Canada
Abstract. The changing character of urban spatial planning over the past two decades is briefly reviewed along with the structures of some of the supporting urban systems models. Most of the available operational urban models are based on the entropy maximizing formalism of Wilson. The weaknesses of this approach are discussed and it is argued that existing models reflect very strongly the geometry of the space over which they are defined. It is emphasized that existing models focus on urban processes and are unable to capture fundamental qualitative changes in urban structure, where these changes may be triggered by changes in the environment or in the processes themselves. Recent attempts to model structural change through applications of catastrophe theory and the theory of dissipative structures are reviewed. Keywords.
Urban systems; models; mathematical analysis; system analysis.
URBAN SPATIAL PLANNING A fundamental distinction is made in this paper between urban structure and urban process. Urban structure, for example, refers to the spatial distributions of residential activity, employment and other human activities. Urban process refers to flows within an urban system such as transport, retail shopping, and so on, and these occur within a particular urban structure. Urban structure is also influenced by the character of the processes themselves. For example, consider the impacts of widespread private car ownership and cheap energy on the structures of North American cities, and the impacts that these structures have had on the transport process. Urban structure and urban process are both influenced by the environment within which they operate. A fundamental policy question in North America at the present time is the probable impact of increasing energy costs on both the transport process and urban structure.
ridors. The emergence of the systems approach in the 1960s shifted the focus of planning to understanding the major urban processes and identifying the key public policy variables that might be manipulated to yield urban spatial structures judged to be socially optimal. This approach to urban spatial planning was supported by a range of large scale computer-based urban systems models. The notions of flexible and adaptive planning began to influence planning practice in the 1970s. Uncertainties about the particular directions of urban development within a changing urban environment led to greater emphasis on flexible and robust urban plans. Great difficulties have been experienced in attempting to adapt and apply the large scale systems models of the 1960s to this new planning thrust. Couclelis [1977) has suggested that the systems typology illustrated in Figure 1 reflects the character of the three planning process variants just described. Type I implies that the interactions between the system and its environment are stable and known and that the concern is with the identification of a systems structure that yields desirable outputs. Type 11 characterizes the systems approach where the behaviour of the system is controlled by a regulator which is part of the environment of the system. Goals for system performance are set and planning is viewed as a search for control policies that will guide the system towards these goals . Type III characterizes the emerging approach to urban spatial planning where the regulator is viewed
During the past two decades the focus of urban spati~l planning has evolved from one concerned with the identification of optimal urban structures, through one preoccupied with the processes and the identification of optimal planning policies, to one concerned primarily with understanding and facilitating structural change. In the 1950s and early 1960s planners were concerned with the preparation of 20-30 year master plans which attempted to create an image of the desired physical structure of a city in terms of major land use types,activity densities and principal transport cor565
566
B. G. Hutchinson and H. M. Couclelis transport facilities, (iv) land servicing decisions, and (v) land use zoning changes.
TYPE 11
TYPE 1
TYltEm
Fig. 1.
Types of Systems
as being part of the system with the internal structure of the system being indefinable to some extent and its future evolution uncertain. Current urban spatial modelling capabilities are confined mainly to the large scale urban systems models. The principal objective of this paper is to review current modelling capabilities and their deficiencies. The second part of the paper reviews recent analytical attempts to model the evolutionary nature of urban systems. URBAN MODELS Putman [1973, 1974], Ayeni [1975 , 1976], Mackett [1976a, 1976b], Echenique [1980], Said and Hutchinson [1980] have all described large scale urban systems models that have been developed as aids to urban strategic planning. While these models do differ in terms of their detailed operations they all have similar conceptual structures. The work of Said and Hutchinson [1980] is briefly reviewed in this paper to illustrate the character of current state-of-the-art models. Figure 2 illustrates the broad structure of this model which consists of the ten basic sub-models identified by the heavier boxes. The state of the urban system at any particular time period is described by (i) the spatial distributions of households stratified by household type, (ii) the spatial distributions of basic, semi-basic and population serving employment stratified by income group, (iii) the spatial distributions of housing stock stratified by dwelling unit type, (iv) the spatial distributions of land use by land use type, and (v) the flows and travel time on each transport link. Exogenously estimated changes in regional activity totals (semi-basic employment, households, housing stock) are allocated spatially as a function of the time t state of the system and the major public policy decisions taken in 6t in order to estimate the state of the system at time t + t>t. Typical public policy decisions taken in ~t might include (i) the locations of new basic employment, (ii) overseas migration policies, (iii) investment in new
The model consists of five activity allocation sub-models (basic employment, semibasic employment, service employment, household, housing stock), four accounting submodels (housing utilization, labour force, job supply, land accounting) to ensure consistency between the sub-models, and a transport sub-model consisting of trip distribution, modal split and equilibrium traffic assignment. The model first calculates the spatial distribution of the expected growth in semi-basic employment for the four semi-basic employment sectors (manufacturing, construction, transportation, wholesale). The inputs to this sub-model are designated vacant industrial land, the employment magnitude in each semibasic employment sector at time t, and the expected total regional growth in each semibasic sector in 6t. The spatial allocation of households by household type is then calculated using as inputs the total employment at time t, the overseas migration rate, and the housing supply at time t and expected during ~t. In the first iteration of the residential location submodel a preliminary estimate of housing stock supply is used which is estimated from the land available for each housing type during ~t and the relative accessibility of each land parcel to employment at t. The spatial allocation of housing stock is calculated using as input the designated residential land, the supply of any public housing and the output of the residential location sub-model. In general there may be a spatial mis-match between the output of the residential location (housing demand) and the housing stock ( housing supply) sub-models. In those zones with unacceptable differences between housing demand and supply in time period 6t, excess demand is re-allocated to zones with available opportunities. The output of the residential location submodel that is obtained after the matching of demand and supply is used to estimate the labour force stratified by income group resident in each zone. The output of the residential location submodel is also used to estimate the new demand for service employment in ~t and this demand is allocated to service centres by the service employment sub-model. This allocation is a function of the attractiveness of the different zones to service demands and the travel times between residential zones and the service employment zones. The total change in the supply of jobs in each zone in ~t is estimated from the sum of the exogenously specified basic employment, and the endogenously estimated semi-basic and service employment allocations.
567
Large Scale Urban Development Models
r.;J~OB;SSU~PPL;;;::yy"jI"L6~tti):-l-----.J WORK TRf'
.----------4-1 SERVICE, SEMI-BASIC
DISTRIBUTION
BASIC
16t )
en
g o CL
SERVICE Df'l.IMotENT
icC
16t) ESTIMATION
AND ALLOCATION
LABOUR FORCE 1l:lt) LAND USE POLICY
16t)
YES
REGIONAL RESIDENTIAr-_ _ _ _ _ _ _ _--t GROWTH I 16 t )
Fig. 2.
Structure of An Urban Systems Model
trip distribution model is used to estimate the new pattern of work trips in At stratified by income group with inputs coming from the spatial distributions of jobs and of labour force and the travel times between residences and work places. Trips to services are estimated in the service employment sub-model and the proportions of these two trip types occurring in the peak period are used to estimate peak vehicle trips.
A
An equilibrium traffic assignment model due to Florian and Nguyen [1975] is used to estimate link volumes and travel times on the transport network. Link travel times produced by the traffic assignment are then used in a minimum path program to produce new estimates of inter-zonal travel times. The new and old travel time matrices are compared and the entire model re-run until the differences between the two matrices are acceptable. Once convergence is achieved between the two sets of travel times a land accounting procedure is performed. The inputs to the land use accounting sub-model include the land uses at t, the changes in activity levels in each zone during At, and a specification of the activities which consume each land use type. The outputs of the land use accounting submodel are the increments in land use consumption of the different land use types and
the total consumptions at t + At. The consumption of each land use type in each zone is subject to capacity constraints. There are two levels of iteration within the model. The first level is within several of the sub-models and these are housing utilization, traffic assignment and land accounting. The second level of iteration is for the entire model where the purpose is to ensure consistency between the travel time matrix estimated from the equilibrium link traffic flows and the ones used to allocate the different activitied whose interactions yield the link flows. Of particular interest to this paper are the nature of the activity allocation functions which are of the following general form: -acij Wje
=
° ,.
i -"-----
.. wje
-ac ij
(1)
j
(2)
where Tij
=
the demand for an activity gen-
ated by zone i and satisfied by opportunities in zone j; 0i= the total demand generated by
B. G. Hutchinson and H. M. Couclelis
568
zone i; W. = the relative attractivity of J
zone j for satisfying the demands; Cij = the travel impedance between zones i and j; a = the sensitivity of trip makers between zones i and j to travel impedances; D. = the amount J
of an activity that is satisfied by the opportunities provided by zone j. Allocation functions of the type defined by equations (1) and (2) were developed by Wilson [1967] using some principles borrowed from statistical mechanics. Urban areas are viewed as closed systems with individual trips or transaction linkages being viewed as the "fundamental particles" of the urban system. Wilson argued that the most likely distribution of demand-supply linkages would be that equivalent to the thermodynamic equilibrium, the entropy maximizing distribution. Prior information about the distribution of linkages at equilibrium is incorporated through the exogenously specified constraints on the distribution. An example of the allocation functions imbedded in the previously described urban systems model is that for allocating households migrating from within Canada :
[La t lk.DUO~(6t)] Y ~ACCE~(t).FiA t
HC~( 6 t)=HC*( t)--~~r.---------~--~----~7
~[~a t lk.DUO(6t)]y t .ACCE~(t).FiA t (3)
with the accessibility to employment function having the following form:
ACCE~
= L Ej.e-a t cij
(4)
t where HC i (6t) = the number of households of type t in 6t; HC~(6t) = the total number of households of type t expected tOI~igrate into the region from Canada in 6t; a t = the probability of a household type t occupying housing stock of type k; DUot(6t) = the number of ty~e k housing stock available in zone i in 6t; Fi t = the proportion of the total households in zone i that are type t ; Ej = the total employment in zone j. Equation (3) is of the typical share form defined by equation (1) with the multi-term multiplicative attractivitytravel deterrence components. The other attraction functions of the urban systems model have a similar form although the particular allocation variables are different. A GEOMETRIC INTERPRETATION OF SPATIAL INTERACTION Cour.lelis 11977] has argued that spatial interaction models of the type just described essentially reflect the geometric space within which they are imbedded rather than any fundamental spatial interaction behaviour. It is argued that the prior information available for model construction and calibration consists of two kinds and these are numerical information obtained by observation and struc-
tural information which is borne by the logico-mathematical structure used to construct models. These abstract structures impart some of their own qualities of atemporal and unconditional validity to the models constructed on their basis. This is part of the reason for the greater or lesser predictive success of mathematical models representing urban processes. Also, because of its independence from empirical fact this a priori structural component of model structures is transferable, not only across space and time, but also from one substantive field to another through the use of analogies. For example, traffic network models borrowed from electrical engineering, urban spatial interaction models based on the gravity law, urban models based on ecological concepts from biology, and so on. Given this perspective then, it is not surprising that interaction models of the type described in equations (1), (2), (3) and (4), containing zonal mass and spatial separation variables, partially "explain" urban spatial interaction patterns. It seems reasonable that the probability of interaction between two urban zones will increase with increasing zone size, and decrease with increasing separation between zones. Urban spatial interaction models may be regarded as analogies which behave in a manner similar to spatial interaction patterns observed with respect to zone systems containing significant amounts of human activities. Similar ideas have been advanced by Batty and March 11976], Cliff, Martin and Ord [1974] and Curry [1972]. Urban gravity models are viewed as a "natural" consequence of the space on which they are defined. The notions of structural prior information and the implications for urban models are developed more fully in Couclelis [1981]. A CONTINGENCY TABLE PERSPECTIVE A complementary view of the notions of structural prior information outlined in the previous section is provided by the work of Willekens [1981] who has suggested that spatial interaction models of the type previously discussed represent special cases of the log-linear model used in contingency table analysis. In the most elementary form, urban transactions of various types incorporated in urban systems models may be displayed as a two-way classification (origins x destinations). In the absence of any other information about the factors influencing the amount of spatial interaction between zones, the maximum likelihood estimates of the interaction magnitudes are given by (5)
where Tij = the expected interaction magnitude or cell count; L* the overall effects given by the geometric mean of all cell counts
Large Scale Urban Development Models reflecting the particular zone system used; Li = the effects due to differences in the marginal frequencies of the origin zones; Lj = the effects due to differences in the marginal frequencies of the destination zones; Lij= differences in interaction magnitudes between zone pairs that reflect particular associations that are not simply due to zone size effects. Equation (5) is subject to the following constraints (6)
ILi = jL j = 1 with the overall effect being given by: L
*
[
T .]l/i j
1T
i ,]. i]
(7)
and the zonal effects by: Li
1
[
L*
1 L. = * ] L
[
1T T .] 1/ j i ,]. i]
1T
i,j
T ]l/i ij
(8)
(9)
Equations (7), (8) and (9) reflect zone size effects and in trip distribution models the Lij term of equation (5) is the travel deterrence term exp(-ac ) defined in equation (1). ij Urban transaction matrices (such as trip tables) for future years are normally estimated by adjusting a prior matrix of spatial interaction structure to new marginal totals using bi-proportional balancing techniques:
569
action terms implied by these tables incorporated in trip distribution and activity allocation models . ESTIMATING STRUCTURAL CHANGE The previous sections of this paper have been concerned with spatial interaction models that attempt to estimate changes in the magnitudes of urban processes for a fixed spatial structure and unchanging urban environment. Harris and Wilson [1978] have described some initial studies of the applications of catastrophe theory to urban spatial interaction models to demonstrate that simple dynamic representations of urban systems may produce qualitative changes in urban spatial structure. They define the well known retail shopping model, a version of equation (1), in the following way: Sij
i3 AiOiWj exp(-ac ij )
A = [E i j
W~ exp(-ac ij ] -1 ]
(11)
(12)
where a and i3 are parameters of the model and the total retail shopping revenue attracted to a zone is given by: (l3)
an4 growth in the retail shopping centres at each location being given by:
(14) (10) where r and Sj = the row and column balancing i factors that ensure that the row and column constraint equations are satisfied; Lij = the prior information on spatial interaction structure incorporated in the model. There are many difficulties with the approach defined in equation (10). For example, Hutchinson and Smith [1979], Sikdar and Hutchinson [1981] have shown that the magnitude of the a-parameter of the travel deterrence function estimated by calibrating distribution models of the type defined in equation (1) is very sensitive to zone size effects. These studies have demonstrated very clearly that spatial interaction models of the type defined by equation (5) are inadequate. A variety of other factors influence spatial interaction magnitudes between zones and these include the characteristics of the housing and job markets, the timing of development, and so on. It seems very clear that the body of theory developed for the analysis of contingency tables must be exploited fully in attempting to develop more consistent and meaningful spatial interaction models. Multidimensional urban transation tables will have to be developed and the higher order inter-
and at equilibrium W. ]
o and (15)
The parameters kj' a and i3 may be viewed as being determined by the environment of the retailing system. kj is proportional to the cost per unit of supplying retail space, i3 reflects the shopping scale economies and diversity available to consumers at larger centres, and a the sensitivity of shoppers to transport costs. Of particular interest is the way in which the parameters of the retail shopping environment influence the retail shopping process Sij' and how these changes in turn influence the structure of the retailing system Wj. Harris and Wilson [1978] have explored the stability of the system and shown the existence of fold catastrophe behaviour in which jumps in behaviour occur as k , a and i3 change. j SELF ORGANIZING SYSTEMS Another approach to this problem of estimating qualitative changes in urban structure is pro-
570
B. G. Hutchinson and H. M. Couclelis
vided by the work of Allen et al [1979, 1980, 1981]. The paradigm underlying this approach is provided by the work of Prigogine and his colleagues in Brussels in the field of physical chemistry. The development of spatial structure is viewed as an evolutionary tree which admits the possibilities of bifurcations occurring due to instabilities in the urban system. Urban systems are viewed as open systems exchanging matter, energy and information with their environments. Branches of the evolutionary tree are viewed as stable expressions of structure and process. The structure may change when it cannot cope with changes induced by changes in its environment or through internal adjustments in the underlying processes. Allen and his colleagues have extended the basic structure of the Lowry-type activity allocation models by writing kinetic equations of the following type for the evolution of the different activities at each location in an urban system: (16)
where Hi = the number of people living in zone i; Ej = the number of jobs located in zone j; Aij = the attractivity of residential zone i as viewed by employees working in zone j where this attractivity measure is defined in a form similar to that described in equation (3). Similar equations are written for the other urban activity sectors. Equation (16) is typical of the so-called ecological models where the rate of growth of an activity in a zone varies with the opportunities still available in that zone. Allen and his colleagues have demonstrated how this approach may produce fundamental changes in urban structure in very simple city systems as model parameters are altered. CONCLUDING REMARKS Available operational urban spatial interaction models are all essentially based on the entropy maximizing paradigm in which equilibrium is equated with the notion of thermodynamic equilibrium. Models of this type have some capability of estimating changes in the magnitudes of urban processes for stable urban structures. However, this predictive ability is related essentially to the structural prior information implicit in the mathematical structures of the models and not to some fundamental spatial interaction behaviour captured by the models. The theory underlying contingency table analysis may be profitably exploited in developing more consistent spatial interaction models. The environment of urban areas is changing rapidly and now analytical techniques are required for exploring the fundamental changes in urban spatial structure that might be triggered by the changing environment. Recent
applications of catastrophe theory and the theory of dissipative structures may be useful in this regard. REFERENCES Allen, P.M., and M. Sanglier (1979). A dynamic model of a central place system. Geographical Analysis, VII, 3, pp. 256272. Allen, P.M. (1980). Planning and decisionmaking in human systems : modelling self-organizaton. Paper presented at NATO Advanced Research Institute on Urban Systems Analysis in Policy Making and Planning, Oxford. Allen, P.M. and M. Sanglier (1981). Urban evolution, self-organization, and decision-making. Environment and Planning A, 13, 2, pp. 167-183. Ayeni, M. (1975). A predictive model of urban stock and activity: 1. theoretical considerations. Environment and Planning A, 7, 8, pp. 965-980. Ayeni~ M. (1976). A predictive model of urban stock and activity: 2. empirical development. Environment and Planning A, 8, 1, pp. 59-78. Batty~ M. and L. March (1976). The method of residues in urban modelling. Environment Planning A, 8, 2, pp. 189-214. Cliff, A.D., R.L. Martin and J. K. Ord (1974). Evaluating the friction of distance parameter in gravity models. Regional Studies, 8, pp. 281-286. Couclelis, H.M. (1977). Urban development models: towards a general theory. Ph.D. thesis. University of Cambridge. Couclelis, H.M. (1981). The function of physical analogies in urban modelling. Transport Group, Department of Civil Engineering, University of Waterloo. Curry, L. (1972). A spatial analysis of gravity flows. Regional Studies 6, pp. 131-147. Echenique, M. (1980). The Sao Paulo Metropolitan Study: A Case Study of the Effectiveness of Urban Systems Analysis. Paper presented at NATO Advanced Research Institute on Urban Systems Analysis in Policy Making and Planning, Oxford. Florian, M. and S. Nguyen, (1975). An application and validation of equilibrium trip assignment methods. Centre de Recherche sur les Transports, Universite de Montreal. Harris, B. and A.G. Wilson (1978). Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models. Environment and Planning A, 10, 4, pp. 371-388. Hutchinson, B.G. and D.P. Smith (1979). The capabilities of the gravity model in explaining journey to work patterns ·in 28 Canadian urban areas. RTAC Forum, 2, 4, pp. 41052. Mackett, R. (1976a). A dynamic integrated activity allocation-transportation model for West Yorkshire, WP40, School
Large Scale Urban Development Models of Geography, University of Leeds. Mackett, R. (1976b). The theoretical structure of a dynamic urban activity stock allocation model, WP135, School of Geography, University of Leeds. Putman, S.H. (1973). The interdependence of transportation development and land development. Department of City and Regional Planning. University of Pennsylvania. Putman, S.H. (1974). Preliminary results from an integrated transportation and land use package, Transportation, 3, pp. 193-224. Said, G.M. and B.G. Hutchinson (1980). An urban systems model for the Toronto region. Paper presented to Roads and Transportation Association of Canada. Sikdar, P.K. and B.G. Hutchinson (1981). Empirical studies of work trip distribution models. Transportation Research, l5A, pp. 233-243. Willekens, F. (1981). Multidimensional population analysis with incomplete data. WP19, Netherlands Inter-University Demographic Institute, Voorburg, The Netherlands. Wilson, A.G. (1967). A statistical theory of spatial interaction models. Transportation Research, 1, pp. 253-269.
571
RECENT EXPERIENCE WITH LARGE SCALE URBAN DEVELOPMENT MODELS B. G. Hutchinson and H. M. Couclelis Department of Civil Engineering, University of Waterloo , Waterloo, Ontario N2L 3Gl, Canada
Abstract. The changing character of urban spatial planning over the past two decades is briefly reviewed along with the structures of some of the supporting urban systems models. Most of the available operational urban models are based on the entropy maximizing formalism of Wilson. The weaknesses of this approach are discussed and it is argued that existing models reflect very strongly the geometry of the space over which they are defined. It is emphasized that existing models focus on urban processes and are unable to capture fundamental qualitative changes in urban structure, where these changes may be triggered by changes in the environment or in the processes themselves. Recent attempts to model structural change through applications of catastrophe theory and the theory of dissipative structures are reviewed. Keywords.
Urban systems; models; mathematical analysis; system analysis.
URBAN SPATIAL PLANNING A fundamental distinction is made in this paper between urban structure and urban process. Urban structure, for example, refers to the spatial distributions of residential activity, employment and other human activities. Urban process refers to flows within an urban system such as transport, retail shopping, and so on, and these occur within a particular urban structure. Urban structure is also influenced by the character of the processes themselves. For example, consider the impacts of widespread private car ownership and cheap energy on the structures of North American cities, and the impacts that these structures have had on the transport process. Urban structure and urban process are both influenced by the environment within which they operate. A fundamental policy question in North America at the present time is the probable impact of increasing energy costs on both the transport process and urban structure.
ridors. The emergence of the systems approach in the 1960s shifted the focus of planning to understanding the major urban processes and identifying the key public policy variables that might be manipulated to yield urban spatial structures judged to be socially optimal. This approach to urban spatial planning was supported by a range of large scale computer-based urban systems models. The notions of flexible and adaptive planning began to influence planning practice in the 1970s. Uncertainties about the particular directions of urban development within a changing urban environment led to greater emphasis on flexible and robust urban plans. Great difficulties have been experienced in attempting to adapt and apply the large scale systems models of the 1960s to this new planning thrust. Couclelis [1977) has suggested that the systems typology illustrated in Figure 1 reflects the character of the three planning process variants just described. Type I implies that the interactions between the system and its environment are stable and known and that the concern is with the identification of a systems structure that yields desirable outputs. Type 11 characterizes the systems approach where the behaviour of the system is controlled by a regulator which is part of the environment of the system. Goals for system performance are set and planning is viewed as a search for control policies that will guide the system towards these goals . Type III characterizes the emerging approach to urban spatial planning where the regulator is viewed
During the past two decades the focus of urban spati~l planning has evolved from one concerned with the identification of optimal urban structures, through one preoccupied with the processes and the identification of optimal planning policies, to one concerned primarily with understanding and facilitating structural change. In the 1950s and early 1960s planners were concerned with the preparation of 20-30 year master plans which attempted to create an image of the desired physical structure of a city in terms of major land use types,activity densities and principal transport cor565
566
B. G. Hutchinson and H. M. Couclelis transport facilities, (iv) land servicing decisions, and (v) land use zoning changes.
TYPE 11
TYPE 1
TYltEm
Fig. 1.
Types of Systems
as being part of the system with the internal structure of the system being indefinable to some extent and its future evolution uncertain. Current urban spatial modelling capabilities are confined mainly to the large scale urban systems models. The principal objective of this paper is to review current modelling capabilities and their deficiencies. The second part of the paper reviews recent analytical attempts to model the evolutionary nature of urban systems. URBAN MODELS Putman [1973, 1974], Ayeni [1975 , 1976], Mackett [1976a, 1976b], Echenique [1980], Said and Hutchinson [1980] have all described large scale urban systems models that have been developed as aids to urban strategic planning. While these models do differ in terms of their detailed operations they all have similar conceptual structures. The work of Said and Hutchinson [1980] is briefly reviewed in this paper to illustrate the character of current state-of-the-art models. Figure 2 illustrates the broad structure of this model which consists of the ten basic sub-models identified by the heavier boxes. The state of the urban system at any particular time period is described by (i) the spatial distributions of households stratified by household type, (ii) the spatial distributions of basic, semi-basic and population serving employment stratified by income group, (iii) the spatial distributions of housing stock stratified by dwelling unit type, (iv) the spatial distributions of land use by land use type, and (v) the flows and travel time on each transport link. Exogenously estimated changes in regional activity totals (semi-basic employment, households, housing stock) are allocated spatially as a function of the time t state of the system and the major public policy decisions taken in 6t in order to estimate the state of the system at time t + t>t. Typical public policy decisions taken in ~t might include (i) the locations of new basic employment, (ii) overseas migration policies, (iii) investment in new
The model consists of five activity allocation sub-models (basic employment, semibasic employment, service employment, household, housing stock), four accounting submodels (housing utilization, labour force, job supply, land accounting) to ensure consistency between the sub-models, and a transport sub-model consisting of trip distribution, modal split and equilibrium traffic assignment. The model first calculates the spatial distribution of the expected growth in semi-basic employment for the four semi-basic employment sectors (manufacturing, construction, transportation, wholesale). The inputs to this sub-model are designated vacant industrial land, the employment magnitude in each semibasic employment sector at time t, and the expected total regional growth in each semibasic sector in 6t. The spatial allocation of households by household type is then calculated using as inputs the total employment at time t, the overseas migration rate, and the housing supply at time t and expected during ~t. In the first iteration of the residential location submodel a preliminary estimate of housing stock supply is used which is estimated from the land available for each housing type during ~t and the relative accessibility of each land parcel to employment at t. The spatial allocation of housing stock is calculated using as input the designated residential land, the supply of any public housing and the output of the residential location sub-model. In general there may be a spatial mis-match between the output of the residential location (housing demand) and the housing stock ( housing supply) sub-models. In those zones with unacceptable differences between housing demand and supply in time period 6t, excess demand is re-allocated to zones with available opportunities. The output of the residential location submodel that is obtained after the matching of demand and supply is used to estimate the labour force stratified by income group resident in each zone. The output of the residential location submodel is also used to estimate the new demand for service employment in ~t and this demand is allocated to service centres by the service employment sub-model. This allocation is a function of the attractiveness of the different zones to service demands and the travel times between residential zones and the service employment zones. The total change in the supply of jobs in each zone in ~t is estimated from the sum of the exogenously specified basic employment, and the endogenously estimated semi-basic and service employment allocations.
567
Large Scale Urban Development Models
r.;J~OB;SSU~PPL;;;::yy"jI"L6~tti):-l-----.J WORK TRf'
.----------4-1 SERVICE, SEMI-BASIC
DISTRIBUTION
BASIC
16t )
en
g o CL
SERVICE Df'l.IMotENT
icC
16t) ESTIMATION
AND ALLOCATION
LABOUR FORCE 1l:lt) LAND USE POLICY
16t)
YES
REGIONAL RESIDENTIAr-_ _ _ _ _ _ _ _--t GROWTH I 16 t )
Fig. 2.
Structure of An Urban Systems Model
trip distribution model is used to estimate the new pattern of work trips in At stratified by income group with inputs coming from the spatial distributions of jobs and of labour force and the travel times between residences and work places. Trips to services are estimated in the service employment sub-model and the proportions of these two trip types occurring in the peak period are used to estimate peak vehicle trips.
A
An equilibrium traffic assignment model due to Florian and Nguyen [1975] is used to estimate link volumes and travel times on the transport network. Link travel times produced by the traffic assignment are then used in a minimum path program to produce new estimates of inter-zonal travel times. The new and old travel time matrices are compared and the entire model re-run until the differences between the two matrices are acceptable. Once convergence is achieved between the two sets of travel times a land accounting procedure is performed. The inputs to the land use accounting sub-model include the land uses at t, the changes in activity levels in each zone during At, and a specification of the activities which consume each land use type. The outputs of the land use accounting submodel are the increments in land use consumption of the different land use types and
the total consumptions at t + At. The consumption of each land use type in each zone is subject to capacity constraints. There are two levels of iteration within the model. The first level is within several of the sub-models and these are housing utilization, traffic assignment and land accounting. The second level of iteration is for the entire model where the purpose is to ensure consistency between the travel time matrix estimated from the equilibrium link traffic flows and the ones used to allocate the different activitied whose interactions yield the link flows. Of particular interest to this paper are the nature of the activity allocation functions which are of the following general form: -acij Wje
=
° ,.
i -"-----
.. wje
-ac ij
(1)
j
(2)
where Tij
=
the demand for an activity gen-
ated by zone i and satisfied by opportunities in zone j; 0i= the total demand generated by
B. G. Hutchinson and H. M. Couclelis
568
zone i; W. = the relative attractivity of J
zone j for satisfying the demands; Cij = the travel impedance between zones i and j; a = the sensitivity of trip makers between zones i and j to travel impedances; D. = the amount J
of an activity that is satisfied by the opportunities provided by zone j. Allocation functions of the type defined by equations (1) and (2) were developed by Wilson [1967] using some principles borrowed from statistical mechanics. Urban areas are viewed as closed systems with individual trips or transaction linkages being viewed as the "fundamental particles" of the urban system. Wilson argued that the most likely distribution of demand-supply linkages would be that equivalent to the thermodynamic equilibrium, the entropy maximizing distribution. Prior information about the distribution of linkages at equilibrium is incorporated through the exogenously specified constraints on the distribution. An example of the allocation functions imbedded in the previously described urban systems model is that for allocating households migrating from within Canada :
[La t lk.DUO~(6t)] Y ~ACCE~(t).FiA t
HC~( 6 t)=HC*( t)--~~r.---------~--~----~7
~[~a t lk.DUO(6t)]y t .ACCE~(t).FiA t (3)
with the accessibility to employment function having the following form:
ACCE~
= L Ej.e-a t cij
(4)
t where HC i (6t) = the number of households of type t in 6t; HC~(6t) = the total number of households of type t expected tOI~igrate into the region from Canada in 6t; a t = the probability of a household type t occupying housing stock of type k; DUot(6t) = the number of ty~e k housing stock available in zone i in 6t; Fi t = the proportion of the total households in zone i that are type t ; Ej = the total employment in zone j. Equation (3) is of the typical share form defined by equation (1) with the multi-term multiplicative attractivitytravel deterrence components. The other attraction functions of the urban systems model have a similar form although the particular allocation variables are different. A GEOMETRIC INTERPRETATION OF SPATIAL INTERACTION Cour.lelis 11977] has argued that spatial interaction models of the type just described essentially reflect the geometric space within which they are imbedded rather than any fundamental spatial interaction behaviour. It is argued that the prior information available for model construction and calibration consists of two kinds and these are numerical information obtained by observation and struc-
tural information which is borne by the logico-mathematical structure used to construct models. These abstract structures impart some of their own qualities of atemporal and unconditional validity to the models constructed on their basis. This is part of the reason for the greater or lesser predictive success of mathematical models representing urban processes. Also, because of its independence from empirical fact this a priori structural component of model structures is transferable, not only across space and time, but also from one substantive field to another through the use of analogies. For example, traffic network models borrowed from electrical engineering, urban spatial interaction models based on the gravity law, urban models based on ecological concepts from biology, and so on. Given this perspective then, it is not surprising that interaction models of the type described in equations (1), (2), (3) and (4), containing zonal mass and spatial separation variables, partially "explain" urban spatial interaction patterns. It seems reasonable that the probability of interaction between two urban zones will increase with increasing zone size, and decrease with increasing separation between zones. Urban spatial interaction models may be regarded as analogies which behave in a manner similar to spatial interaction patterns observed with respect to zone systems containing significant amounts of human activities. Similar ideas have been advanced by Batty and March 11976], Cliff, Martin and Ord [1974] and Curry [1972]. Urban gravity models are viewed as a "natural" consequence of the space on which they are defined. The notions of structural prior information and the implications for urban models are developed more fully in Couclelis [1981]. A CONTINGENCY TABLE PERSPECTIVE A complementary view of the notions of structural prior information outlined in the previous section is provided by the work of Willekens [1981] who has suggested that spatial interaction models of the type previously discussed represent special cases of the log-linear model used in contingency table analysis. In the most elementary form, urban transactions of various types incorporated in urban systems models may be displayed as a two-way classification (origins x destinations). In the absence of any other information about the factors influencing the amount of spatial interaction between zones, the maximum likelihood estimates of the interaction magnitudes are given by (5)
where Tij = the expected interaction magnitude or cell count; L* the overall effects given by the geometric mean of all cell counts
Large Scale Urban Development Models reflecting the particular zone system used; Li = the effects due to differences in the marginal frequencies of the origin zones; Lj = the effects due to differences in the marginal frequencies of the destination zones; Lij= differences in interaction magnitudes between zone pairs that reflect particular associations that are not simply due to zone size effects. Equation (5) is subject to the following constraints (6)
ILi = jL j = 1 with the overall effect being given by: L
*
[
T .]l/i j
1T
i ,]. i]
(7)
and the zonal effects by: Li
1
[
L*
1 L. = * ] L
[
1T T .] 1/ j i ,]. i]
1T
i,j
T ]l/i ij
(8)
(9)
Equations (7), (8) and (9) reflect zone size effects and in trip distribution models the Lij term of equation (5) is the travel deterrence term exp(-ac ) defined in equation (1). ij Urban transaction matrices (such as trip tables) for future years are normally estimated by adjusting a prior matrix of spatial interaction structure to new marginal totals using bi-proportional balancing techniques:
569
action terms implied by these tables incorporated in trip distribution and activity allocation models . ESTIMATING STRUCTURAL CHANGE The previous sections of this paper have been concerned with spatial interaction models that attempt to estimate changes in the magnitudes of urban processes for a fixed spatial structure and unchanging urban environment. Harris and Wilson [1978] have described some initial studies of the applications of catastrophe theory to urban spatial interaction models to demonstrate that simple dynamic representations of urban systems may produce qualitative changes in urban spatial structure. They define the well known retail shopping model, a version of equation (1), in the following way: Sij
i3 AiOiWj exp(-ac ij )
A = [E i j
W~ exp(-ac ij ] -1 ]
(11)
(12)
where a and i3 are parameters of the model and the total retail shopping revenue attracted to a zone is given by: (l3)
an4 growth in the retail shopping centres at each location being given by:
(14) (10) where r and Sj = the row and column balancing i factors that ensure that the row and column constraint equations are satisfied; Lij = the prior information on spatial interaction structure incorporated in the model. There are many difficulties with the approach defined in equation (10). For example, Hutchinson and Smith [1979], Sikdar and Hutchinson [1981] have shown that the magnitude of the a-parameter of the travel deterrence function estimated by calibrating distribution models of the type defined in equation (1) is very sensitive to zone size effects. These studies have demonstrated very clearly that spatial interaction models of the type defined by equation (5) are inadequate. A variety of other factors influence spatial interaction magnitudes between zones and these include the characteristics of the housing and job markets, the timing of development, and so on. It seems very clear that the body of theory developed for the analysis of contingency tables must be exploited fully in attempting to develop more consistent and meaningful spatial interaction models. Multidimensional urban transation tables will have to be developed and the higher order inter-
and at equilibrium W. ]
o and (15)
The parameters kj' a and i3 may be viewed as being determined by the environment of the retailing system. kj is proportional to the cost per unit of supplying retail space, i3 reflects the shopping scale economies and diversity available to consumers at larger centres, and a the sensitivity of shoppers to transport costs. Of particular interest is the way in which the parameters of the retail shopping environment influence the retail shopping process Sij' and how these changes in turn influence the structure of the retailing system Wj. Harris and Wilson [1978] have explored the stability of the system and shown the existence of fold catastrophe behaviour in which jumps in behaviour occur as k , a and i3 change. j SELF ORGANIZING SYSTEMS Another approach to this problem of estimating qualitative changes in urban structure is pro-
570
B. G. Hutchinson and H. M. Couclelis
vided by the work of Allen et al [1979, 1980, 1981]. The paradigm underlying this approach is provided by the work of Prigogine and his colleagues in Brussels in the field of physical chemistry. The development of spatial structure is viewed as an evolutionary tree which admits the possibilities of bifurcations occurring due to instabilities in the urban system. Urban systems are viewed as open systems exchanging matter, energy and information with their environments. Branches of the evolutionary tree are viewed as stable expressions of structure and process. The structure may change when it cannot cope with changes induced by changes in its environment or through internal adjustments in the underlying processes. Allen and his colleagues have extended the basic structure of the Lowry-type activity allocation models by writing kinetic equations of the following type for the evolution of the different activities at each location in an urban system: (16)
where Hi = the number of people living in zone i; Ej = the number of jobs located in zone j; Aij = the attractivity of residential zone i as viewed by employees working in zone j where this attractivity measure is defined in a form similar to that described in equation (3). Similar equations are written for the other urban activity sectors. Equation (16) is typical of the so-called ecological models where the rate of growth of an activity in a zone varies with the opportunities still available in that zone. Allen and his colleagues have demonstrated how this approach may produce fundamental changes in urban structure in very simple city systems as model parameters are altered. CONCLUDING REMARKS Available operational urban spatial interaction models are all essentially based on the entropy maximizing paradigm in which equilibrium is equated with the notion of thermodynamic equilibrium. Models of this type have some capability of estimating changes in the magnitudes of urban processes for stable urban structures. However, this predictive ability is related essentially to the structural prior information implicit in the mathematical structures of the models and not to some fundamental spatial interaction behaviour captured by the models. The theory underlying contingency table analysis may be profitably exploited in developing more consistent spatial interaction models. The environment of urban areas is changing rapidly and now analytical techniques are required for exploring the fundamental changes in urban spatial structure that might be triggered by the changing environment. Recent
applications of catastrophe theory and the theory of dissipative structures may be useful in this regard. REFERENCES Allen, P.M., and M. Sanglier (1979). A dynamic model of a central place system. Geographical Analysis, VII, 3, pp. 256272. Allen, P.M. (1980). Planning and decisionmaking in human systems : modelling self-organizaton. Paper presented at NATO Advanced Research Institute on Urban Systems Analysis in Policy Making and Planning, Oxford. Allen, P.M. and M. Sanglier (1981). Urban evolution, self-organization, and decision-making. Environment and Planning A, 13, 2, pp. 167-183. Ayeni, M. (1975). A predictive model of urban stock and activity: 1. theoretical considerations. Environment and Planning A, 7, 8, pp. 965-980. Ayeni~ M. (1976). A predictive model of urban stock and activity: 2. empirical development. Environment and Planning A, 8, 1, pp. 59-78. Batty~ M. and L. March (1976). The method of residues in urban modelling. Environment Planning A, 8, 2, pp. 189-214. Cliff, A.D., R.L. Martin and J. K. Ord (1974). Evaluating the friction of distance parameter in gravity models. Regional Studies, 8, pp. 281-286. Couclelis, H.M. (1977). Urban development models: towards a general theory. Ph.D. thesis. University of Cambridge. Couclelis, H.M. (1981). The function of physical analogies in urban modelling. Transport Group, Department of Civil Engineering, University of Waterloo. Curry, L. (1972). A spatial analysis of gravity flows. Regional Studies 6, pp. 131-147. Echenique, M. (1980). The Sao Paulo Metropolitan Study: A Case Study of the Effectiveness of Urban Systems Analysis. Paper presented at NATO Advanced Research Institute on Urban Systems Analysis in Policy Making and Planning, Oxford. Florian, M. and S. Nguyen, (1975). An application and validation of equilibrium trip assignment methods. Centre de Recherche sur les Transports, Universite de Montreal. Harris, B. and A.G. Wilson (1978). Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models. Environment and Planning A, 10, 4, pp. 371-388. Hutchinson, B.G. and D.P. Smith (1979). The capabilities of the gravity model in explaining journey to work patterns ·in 28 Canadian urban areas. RTAC Forum, 2, 4, pp. 41052. Mackett, R. (1976a). A dynamic integrated activity allocation-transportation model for West Yorkshire, WP40, School
Large Scale Urban Development Models of Geography, University of Leeds. Mackett, R. (1976b). The theoretical structure of a dynamic urban activity stock allocation model, WP135, School of Geography, University of Leeds. Putman, S.H. (1973). The interdependence of transportation development and land development. Department of City and Regional Planning. University of Pennsylvania. Putman, S.H. (1974). Preliminary results from an integrated transportation and land use package, Transportation, 3, pp. 193-224. Said, G.M. and B.G. Hutchinson (1980). An urban systems model for the Toronto region. Paper presented to Roads and Transportation Association of Canada. Sikdar, P.K. and B.G. Hutchinson (1981). Empirical studies of work trip distribution models. Transportation Research, l5A, pp. 233-243. Willekens, F. (1981). Multidimensional population analysis with incomplete data. WP19, Netherlands Inter-University Demographic Institute, Voorburg, The Netherlands. Wilson, A.G. (1967). A statistical theory of spatial interaction models. Transportation Research, 1, pp. 253-269.
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